1. Field of the Invention
This invention relates to the improved determination of the distribution and intensity of signal sources of initially imprecisely specified spatial amplitude and/or temporal characteristics in the presence of noise and distortion by means of appropriate measurement and analysis.
2. Description of the Prior Art
In many communication applications it is desired to extract information of known frequency, phase, or repetitive characteristics from a noisy signal. Many techniques have been developed to successfully discriminate against essentially random noise and so dramatically improve the signal to noise ratio. In radioisotope measurements and many other applications, however, the signal is itself intrinsically randomly fluctuating, and many of the communications techniques are not immediately applicable. Furthermore in, for example, radioisotopic organ scans the information desired is typically three-dimensional (four-dimensional if time-dependent dynamic functions are being studied), while the measured outputs are often only two-dimensional scans. Specialized equipment and methods of data analysis have therefore been developed in the field of radioisotopic medical imaging, such as described in "Radionuclide Section Scanning," by Ell et al, Grune & Stratton (1982). Radioisotope tissue distributions have been determined using positron emission coincidence techniques. These methods are currently limited as to the organ scanned, the available positron emitting isotopes and require specialized equipment usually with either 2.pi. planar or .apprxeq.4.pi. solid angle geometries. Attempts have also been made to reconstruct three-dimensional radioisotopic tissue distributions by acquiring two-dimensional signal gamma scan data at different angles of projection and carrying out a three-dimensional reconstruction by projective techniques, such as described in "Image Reconstruction From Projections," by Herman (1980, Academic Press). Here, too, specialized equipment usually with 2.pi. planar or .apprxeq.4.pi. solid angle geometries were employed.
In general, the mathematical difficulties associated with a formal linear algebraic approach to the problem of determining the three-dimensional tissue distributions from measured data are very formidable. Formal solutions would not only involve inversion of a series of relatively large three-dimensional point source response matrices, but the utilization of noisy data appropriately weighted in accordance with overall statistical optimization constraints. In view of these difficulties, it has been customary instead to use various iterative approximation techniques, generally making use of projective data at multiple angles of orientation. (It should be noted, however, that if the point source response is itself sufficiently depth dependent, there is, in principle, no a priori requirement that multiple projections be taken.)
These iterative approximation techniques have been variably successful in extracting the true source distribution from the convoluted noisy data. Recently an approach has been developed by Shepp and Vardi (1982, IEEE Tran. Medical Imaging), and Lange and Carson (1984, J. Comput. Ass. Tomography) which makes use of modern statistical methodology to derive algorithms which under certain conditions serve to formally specify those source distributions which are probabilistically most likely, considering the measured data, the statistical distributions satisfied by the data (e.g. Poisson distribution, or Gauss distribution) and the known point source response function (PSRF, distortion characteristics of the equipment and measuring process). Excellent solutions are achieved if the PSRF are sharp implying relatively little distortion to begin with and in the presence of minimal noise. However, these maximum likelihood algorithms (ML) do not in general give accurate results when the PSRF is broad and the data is noisy. Since a broad PSRF implies a low resolution detection system and noisy data is necessarily inherently less reliable, the limitations in solution specificity would appear to be inherent and the above approaches therefore still optimal.
Nonetheless in certain situations of very basic importance, a further major improvement in image processing can be achieved.
Consider a situation, for example, in which while the spatial distribution of the source distribution is relatively unknown, the radioisotope concentration of the individual source elements is readily estimated. This occurs, for example, in cardiac imaging immediately following I.V. administration of a radioisotopic dose. The blood concentration curves can be relatively well measured or otherwise estimated over the next few minutes while the cardiac muscle will be comparatively free of isotope.
The prior ML algorithms in specifying solutions which do not take into account such source information are, therefore, not actually extracting the general maximum likelihood result (i.e. that maximum likelihood source distribution considering all of the available information).
It follows then that there has not been a generally effective method for extracting the most probable source distribution which takes into account all of the information which may be available.
These observations can be expressed more precisely in terms of the standard statistical relation. Bayes' Law: EQU P(.PHI..vertline.N)=P(N.vertline..PHI.)P(.PHI.)/P(N) (1)
where:
P(.PHI..vertline.N) is the conditional probability of the source vector .PHI. subject to the data vector N; PA1 P(N.vertline..PHI.) is the conditional probability of the data vector N subject to the source vector .PHI.; PA1 P(.PHI.) is the probability distribution of the source vector .PHI. (assumed competely unknown or constant in the Shepp and Vardi, and Lange and Carson maximum likelihood formulations but not in the new Bayesian formalism); and PA1 P(N) is the probability distribution of the data vector N (treated as a constant in all methods).
When P(.PHI.) and P(N) are assumed constant, P(.PHI..vertline.N) and P(N.vertline..PHI.) are of course linearly proportional. Maximizing P(N.vertline..PHI.) (which is the approach previously taken) is then equivalent to maximizing P(.PHI..vertline.N). In the new Bayesian formalism, however, any non-trivial a priori source information is first incorporated in P(.PHI.) and the more logical direct maximization of P(.PHI..vertline.N) carried out, taking into account, of course, the now non-constant P(.PHI.) on the right side of Eq.(1).